RELATIVISTIC TRANSFORMATIONS FOR TIME SYNCHRONIZATION AND

DISSEMINATION IN THE SOLAR SYSTEM

Robert A. NelsonSatellite Engineering Research Corporation

7701 Woodmont Avenue, Suite 208, Bethesda, MD 20814 USA

RobtNelson@aol.com

International Astronomical Union XXVIth General AssemblyPrague, Czech Republic

Commission 31 (Time)Monday, August 21, 2006

Time is essential to navigation

H1 (1730-35) H2 (1737-40) H3 (1740-55) H4 (1755-59) H5 (1772)

John Harrison solved the problem of determining longitude at sea with the invention of the marine chronometer.

A replica of H4 made by Larcum Kendall was carried by Captain James Cook in his voyage of discovery to the South Pacific in 1772.

Need for relativistic algorithms

On the surface of the Earth, time is required to determine longitude.

Similarly, for future navigation in space, a coherent time reference will be required.

For navigation at the one-meter level of precision, the time in a well defined coordinate system at a level of the order 10 ns is needed.

Mathematical algorithms taking into account the principles of general relativity are needed for consistent synchronization and dissemination of time.

The Global Positioning System (GPS) provides a paradigm for the appropriate treatment of relativity in a practical engineering system.

In the theory of general relativity, there are two kinds of time.

Proper time is the actual reading of a clock. The proper times are different for clocks in different gravitational potentials and in different states of motion.

The proper time measured by a clock may be compared to the proper time measured by another clock through the intermediate variable called coordinate time.

Coordinate time, by definition, has the same value everywhere for a given event.

The relationship between proper time and coordinate time is established through the invariance of the four-dimensional space-time interval.

Proper time and coordinate time

Invariant space-time interval

3 3 3 3 32 2 2

00 00 0 1 1 1

2 j i jj i j

j i j

ds g dx dx g c dt g c dt dx g dx dx

3 32 2 2

0 0

ds g dx dx c d

3 32

0 0

0ds g dx dx

The theory of space, time, and gravitation according to the general theory of relativity is founded upon the notion of an invariant Riemannian space-time interval of the form

For a transported clock, the space-time interval is

For an electromagnetic signal, the space-time interval satisfies the condition

Earth-orbiting satellite clocks

g00 1 – 2 U / c2 , g0 j = 0, and gi j  i j

0

22 2

1 1 11

2t U v d

c c

0

20 02 2 2

1 1 1 11 1 ( )

2t W t U W v d

c c c

02 2 2

3 1 1 21 sin

2

GMt W GM a e E

c a c c

2 2

2 2sinrelt GM a e E

c c

r v

To a sufficient approximation in the analysis of clock transport, the components of the metric tensor in an Earth-Centered Inertial (ECI) coordinate system are

The elapsed coordinate time with respect to an ECI frame of reference is

It is convenient to apply a change of scale

The elapsed coordinate time for an Earth-orbiting satellite clock becomes

The second term due to the orbital eccentricity may be expressed

Electromagnetic signals from Earth-orbiting clocks

 g00 1 , g 0 j = (  r) j / c , and g i j    i j 

30

1 00path

1 j i

j

gt dx

c c g

Sagnac 2 2 2 2

1 1 1 22

B B B

A A A

At d d d

c c c c

ω r r ω r r ω A

Sagnac 2 2

2A B A B

At x y y x

c c

The coordinate time of propagation of an electromagnetic signal is

In the rotating ECEF coordinate system, the metric components are

The integral term is called the Sagnac effect, which may be expressed.

For endpoints at (xA , yA) and (xB , yB), the Sagnac effect is

The Global Positioning System (GPS)

The GPS provides a paradigm for the application of the principles of general relativity to a practical engineering system.

The fundamental measurement is pseudorange (PR), which is the phase shift necessary to bring about correlation between a pseudorandom noise (PRN) signal transmitted by the satellite and a replica PRN signal generated in the receiver.

The PR measurements must be corrected for ionospheric delay, tropospheric delay, satellite clock offset, and relativity. Four PR measurements are required to solve for x, y, z, and T.

Relativistic effects in the GPS

For measurements with a precision at the one‑to‑ten nanosecond level, there are three relativistic effects that must be considered.

First, there is the effect of time dilation. GPS satellites revolve around the Earth with an orbital period of 11.967 hours and a velocity of 3.874 km/s. On account of its velocity, a GPS satellite clock appears to run slow by 7 s per day.

Second, there is the effect of the gravitational redshift. At an altitude of 20 184 km, the clock runs fast by 45 s per day.

The net effect of time dilation and gravitational redshift is that the satellite clock runs fast by approximately 38 s per day when compared to a similar clock at rest on the geoid, including the effects of the Earth’s rotation and the gravitational potential at the Earth’s surface.

With an orbital eccentricity of 0.02, there is a relativistic sinusoidal variation in the apparent clock time having an amplitude of 46 ns at the orbital period.

The third relativistic effect is the Sagnac effect. For a stationary terrestrial receiver on the geoid, the Sagnac correction can be as large as 133 ns.

Relativistic Effects in

Earth-Orbiting Satellite Clocks

Constants

Velocity of light m/s 299 792 458

Gravitational constant of Earth km3/s2 398 600.44

Radius of Earth km 6378.137

J2 oblateness coefficient 0.0010826

Angular velocity of Earth rotation rad/s 7.292 10 5

Geopotential on geoid U0 m2/s2 6.264 107

U0/c2 -6.969 10 10

Satellite orbital properties

Satellite ISS GLONASS GPS Galileo Molniya GEO

Semimajor axis km 6766 25510 26561.8 29994 26562 42164

Eccentricity 0.01 0.02 0.02 0.02 0.722 0.01

Inclination deg 51.6 64.8 55.0 56.0 63.4 0.1

Argument of perigee deg 0 0 0 0 250 0

Apogee altitude km 456 19642 20715 24216 39362 36208

Perigee altitude km 320 18622 19652 23016 1006 35364

Ascending node altitude km 320 18622 19652 23016 10507 35364

Period of revolution s 5539 40549 43082 51697 43083 86164

Mean motion rev/d 15.6 2.1 2.0 1.7 2.0 1.0

Mean velocity km/s 7.675 3.953 3.874 3.645 3.874 3.075

Clock effects

Secular time dilation s/d -28.2 -7.4 -7.1 -6.3 -7.1 -4.4

Secular redshift s/d 3.5 45.1 45.7 47.3 45.7 51.0

Net secular effect s/d -24.7 37.7 38.6 41.1 38.6 46.6 Amplitude of periodic effect due to eccentricity ns 12 45 46 49 1653 29 Secular oblateness contribution to redshift ns/d 23.7 0.8 0.5 0.4 2.5 -0.1 Amplitude of periodic effect due to oblateness ps 264 50 38 33 167 0 Amplitude of periodic tidal effect of Moon ps 0.0 1.0 1.2 1.8 1.2 6.1 Amplitude of periodic tidal effect of Sun ps 0.0 0.5 0.5 0.8 0.5 2.7

Signal propagation

Maximum Sagnac effect ns 13 131 136 155 234 218 Gravitational propagation delay along radius ps 0.8 -3.5 -4.7 -9.1 -4.7 -27.3 Amplitude of periodic fractional Doppler shift 10-12 13.1 7.0 6.7 5.9 241.1 2.1

Time transfer between Mars and Earth

Proper time as measured by clock on Mars spacecraft

Proper time as measured by clocks on Mars surface

Barycentric Coordinate Time (TCB

Proper time as measured by clocks on Earth’s surface

Terrestrial Time (TT)

International Atomic Time (TAI)

Coordinated Universal Time (UTC)

GPS Time

Earth

Mars Spacecraft

Sun

Mars

Proper time as measured by clock on GPS satellite

GPS Satellite

Terrestrial Time (TT) and Barycentric Coordinate Time (TCB)

TT TAI 32.184 s

20TCG TT ( / )E GW c T L T

0

22 2

1 1 11 ( )

2t U v d

c c

r

Terrestrial Time (TT) is a proper time realized by clocks on the geoid. A practical realization of TT in terms of International Atomic Time (TAI) is

The transformation from TT to Geocentric Coordinate Time (TCG) is

where W0E is Earth geopotential, LG W0E / c2 = 6.969 290 134  1010 60.2 s/d, and T is the time elapsed since 1 January 1977 0 h TAI (JD 244 3144.5).

Barycentric Coordinate Time (TCB) is the coordinate time t in a barycentric coordinate system. The coordinate time t corresponding to the proper time maintained by a clock on the geoid is

Modification of integral for coordinate time

E r r R

E v v R

ext ext ext( ) ( )EU U U r r R

2 2 22 | |E Ev v R v R

( )E E E

d

dt R v R v R a

extE

E

dU

dt

va

0 0 0

2 2ext2 2 2

1 1 1 1 1| |

2 2

t t t

E E E Et t t

t U v dt U dtc c c

r R R R v

U(r) = UE(r) + Uext(r)

It is desirable to separate the clock–dependent part from the clock–independent part.

Thus the elapsed coordinate time is

Barycentric Coordinate Time - Terrestrial Time

0 0

0

2ext2 2 2

1 1 1 1TCB TT =

2

tt t

E E G E C G Et tt

U v dt L T L T P L Tc c c

r R v R v

0

2 2 2 2

1 1 3 1 2sin

2 2

t

SE ext E E S E E E

t E

GMU v dt T GM a e E

c c a c

r

where LC = 1.480 826 867 41  108 1.28 ms/d and T is the time elapsed since 1 January 1977 0 h TAI (JD 2443144.5). For a clock on the geoid, the diurnal term has a maximum amplitude of 2.1 s (for a clock on the equator). The leading terms in the evaluation of the integral are

The proper time is identified with Terrestrial Time (TT). The coordinate time t is identified with Barycentric Coordinate Time (TCB). Thus

Barycentric Coordinate Time - Mars Time

0 0

0

2ext2 2 2

1 1 1 1TCB MT =

2

tt t

M M M M CM M Mt tt

U v dt L T L T P L Tc c c

r R v R v

0

2 2 2 2

1 1 3 1 2sin

2 2

t

SM ext M M S M M M

t M

GMU v dt T GM a e E

c c a c

r

Similar transformations apply to the transfer of time from the surface of Mars to the solar system barycenter.

The transformation from Mars Time (MT) to Areocentric Coordinate Time (TCA) is

Then the transformation between TCB and MT is

where

TCA – MT = (W0M / c2) T = LM T

Geoid to geocenter Secular drift 60.2 s/d Maximum amplitude of diurnal term 2.1 s

Geocenter to Barycenter Secular drift 1.28 ms/d Amplitude of principal periodic term 1.7 ms

Mars surface to Mars center Secular drift 12.1 s/d Maximum amplitude of diurnal term 0.9 s

Mars center to Barycenter Secular drift 0.84 ms/d Amplitude of principal periodic term 11.4 ms

Summary of relativistic effects

MT – TT = (TCB – TT) – (TCB – MT)

The net secular drift is (1.28 ms/d + 0.06 ms/d) – (0.84 ms/d + 0.01 ms/d) = 0.49 ms/d.

Net relativistic periodic effect for time transfer between Mars and Earth

250 500 750 1000 1250 1500 1750

-10

-5

5

10

ms

days

Relativistic Transformation from the Moon to Earth

TCG TT GL T

0 0

0

2ext2 2 2

1 1 1 1TCB LT =

2

tt t

m m m m Cm m mt tt

U v dt L T L T P L Tc c c

r R v R v

2

3 1

2E

Cmm

GML

c a

2

2sinE m m mP GM a e E

c 2

1 mm

m

GML

c R

LT – TT = (TCG – TT) – (TCG – LT)

Thus the net secular drift rate is 60.2 s/d – (1.5 s/d + 2.7 s/d) = 56.0 s/d and the amplitude of the periodic effect is 0.48 s at the Moon’s orbital period (27.3 d).

For time transfer from the surface of the Moon to the surface of the Earth, the procedure is similar but the relative magnitudes of the terms is different. A convenient coordinate system is one whose origin is at the center of the Earth. (The motion of the Earth’s center about the center of mass of the Earth‑Moon system will be neglected.) The difference between Geocentric Coordinate Time and Terrestrial Time, the proper time measured by clocks on the Earth’s surface, is

Also, the difference between Geocentric Coordinate Time (TCG) and Lunar Time (LT), the proper time measured by clocks on the Moon’s surface, is

Conclusion

In the future exploration of the solar system, there will be a need for coherent time synchronization and dissemination. This will require the application of appropriate relativistic algorithms in a common coordinate system (such as the barycentric coordinate system).

The effects of relativity are by no means negligible. They have been successfully demonstrated in the Global Positioning System (GPS). Analogous algorithms will be required in the future exploration of the Moon, Mars, and beyond.

The amplitude of the relativistic difference between the proper time reading of a clock on Mars and the proper time reading of a clock on Earth is up to 13.1 ms. Neglect of this effect using GPS-like signals transmitted by clocks on Mars and an ephemeris expressed in terms of Terrestrial Time would cause a navigation error of about 3000 km.

Appendix

Planetary Reference Data

Reference Data for the Earth and Mars

MassSun 1.9891 1030 kgEarth 5.9742 1024 kgMars 0.6419 1024 kg

Planetary radiusEarth 6378 kmMars 3397 km

Orbital semimajor axisEarth 1.000 AU = 1.496 108 kmMars 1.524 AU = 2.279 108 km

Orbital periodEarth 365.2422 dMars 686.9297 d

Average orbital velocityEarth 29.8 km/sMars 24.1 km/s

Orbital eccentricityEarth 0.0167Mars 0.0934

Speed of light 299,792,458 m/s (exact)Gravitational constant 6.6726 1011m3 / kg s2

Reference Data for the Moon

Mass 0.07353 1024 kg

Radius 1738.2 kmOrbital semimajor axis 384,400 kmOrbital eccentricity 0.05490Average orbital velocity 1.023 km/sDistance of geocenter from barycenter 4671 km